Mesh simplification is a well studied field with long history of research. A mesh can be simplified progressively by two techniques, namely vertex decimation and edge collapse. Although the first one is more interpretable, it requires re-tessellation in order to fill the generated holes. In the general case, each vertex is assigned with an importance score, which may indicate its distance from the average plane [1]}, its perceptual importance [2]}, or high curvature [3]}, ensuring that vertices at smooth regions will be decimated before vertices with sharp features. Edge folding was first introduced in the seminal work of [4]}, where several transforms such as edge swap, edge split and edge collapse were introduced to minimize a simplification energy function. [5]} associated each vertex with the set of planes in its 1-hop neighborhood and defined the edge collapse cost as the maximum distance between the resulting vertex and the collapsed points' planes. [6]} observed that each plane can be expressed by a fundamental quadric matrix and they utilized an additive rule to express the distance of a point from the respective set of planes, i.e., the distance of a point from a set of planes can be expressed using the sum of their quadrics. Based on this observation, the authors proposed a Quadric Error Metric (QEM) to measure the error introduced when an edge collapses and simplified the edges with the minimum error. In a follow up work [7]}, the authors generalized the quadric matrix to handle high dimensional attributes. [8]} observed that greedy simplification methods lead to sub-optimal meshes and attempted to tackle mesh simplification as a global optimization problem using shape proxies. In particular, they introduced a normal deviation error metric to partition the input mesh to non-overlapping connected regions and then fit plane approximations (shape proxies) to each partition. Although the process produces more accurate shape approximations of the input, the method is not particularly efficient.
